Lindblad Operator Computation for a Single-Mode Quantum Field in a Cylindrical Fibre Based on the Boltzmann Equation to Account for Random Scattering by the Phonon Lattice

The first and second order Boltzmann kinetic transport equations, taking into account external and internal electromagnetic interactions of the charged phonon lattice, are set up. The incident classical field component interacts with the phonon lattice and gets scattered randomly owing to the random motion of the phonons. The incident quantum field component thus interacts with both the incident classical field and the scattered classical field, thereby generating respectively non-random and random Hamiltonian perturbations to the total quantum field Hamiltonian. By using a second order Taylor expansion of the mixed state Schrödinger evolution of the quantum field, taking into account these nonrandom and random perturbations, and then forming statistical averages for the random component using the first and second order Boltzmann distributions of the particles in phase space, we are able to calculate the Lindblad operator term corresponding to the random component. We then demonstrate how to cancel out these nonrandom and Lindblad perturbations to the mixed state dynamics using counterterms, based on a Monte Carlo method involving generating a sequence of iid random Hamiltonian operators, applying Schrödinger evolution, and then forming ensemble averages making use of the strong law of large numbers.